You place a penny on a turntable at a distance of 10.0 cm from the center. The coefficient of st

You place a penny on a turntable at a distance of 10.0 cm...

You place a penny on a turntable at a distance of $10.0 \mathrm{cm}$ from the center. The coefficient of static friction between the penny and the turntable is $0.350 .$ The turntable's angular acceleration is $2.00 \mathrm{rad} / \mathrm{s}^{2} .$ How long after you turn on the turntable will the penny begin to slide off of the turntable?

Key Concepts

Static Friction

Static friction is the force that resists the initiation of motion between two surfaces in contact. For a rotating object, static friction must provide the necessary centripetal force to keep the object moving in a circle. The maximum static friction available is given by ?_s times the normal force, and once the needed centripetal force exceeds this, the object will begin to slide.

Rotational Kinematics

Rotational kinematics deals with the motion of objects that rotate about an axis, analogous to linear kinematics. Using equations such as ? = ?? + ?t allows us to determine the angular velocity at any time, where ? is the constant angular acceleration. This calculated angular velocity is then used to determine the centripetal acceleration acting on the object.

Centripetal Acceleration

In circular motion, any object moving in a circle must continuously change direction. This requires a centripetal acceleration directed toward the center of the circle, defined by the equation a = ?²r, where ? is the angular velocity and r is the radius of the circle. This acceleration is crucial in understanding when a frictional force is needed to maintain circular motion.

Transcript

00:01 In this problem, we will be investigating how fast a turntable has to get going to make it so that a penny sitting on the surface of the turntable starts slipping off.

00:14 So our turntable here is going to be accelerating at a constant angular acceleration of two radians per second squared.

00:24 The coefficient of static friction between the turntable and the penny will be 0 .35.

00:30 And the penny will be sitting 10 centimeters away from the center of the turntable.

00:36 So the question is, at what point will the penny actually start slipping? and that point is when the force of friction, the force of static friction, is overcome by what the radial force needs to be.

00:59 So what i mean by that is if we look at a free body diagram of our penny, in order to stay in this circular orbit, this way, it needs to have a radial force inward like this, fr.

01:27 And that force is going to be the force of friction.

01:35 But at a certain point, the force of static friction can only get so large depending on the normal force of the penny on this turntable.

01:43 And once that is breached, the penny will no longer be able to stay put and it will start moving.

01:52 So we can write this out mathematically.

01:56 We can say that that radial force, m omega squared r, whenever it is greater or equal to the force of friction is when our penny will start slipping.

02:21 Now remember this force of static friction here will be massed.

02:26 Maximized when it is equal to mu times the normal force...

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